| Popis: |
Let $${\rm{\backslash mathscr\{ C\} }}$$ be a triangulated category and $${\rm{\backslash mathscr\{ X\} }}$$ be a cluster tilting subcategory of $${\rm{\backslash mathscr\{ C\} }}$$ . Koenig and Zhu showed that the quotient category $${\rm{\backslash mathscr\{ C\} / \backslash mathscr\{ X\} }}$$ is Gorenstein of Gorenstein dimension at most one. But this is not always true when $${\rm{\backslash mathscr\{ C\} }}$$ becomes an exact category. The notion of an extriangulated category was introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. Now let $${\rm{\backslash mathscr\{ C\} }}$$ be an extriangulated category with enough projectives and enough injectives, and $${\rm{\backslash mathscr\{ X\} }}$$ a cluster tilting subcategory of $${\rm{\backslash mathscr\{ C\} }}$$ . We show that under certain conditions, the quotient category $${\rm{\backslash mathscr\{ C\} / \backslash mathscr\{ X\} }}$$ is Gorenstein of Gorenstein dimension at most one. As an application, this result generalizes the work by Koenig and Zhu. |
